This invention relates to the field of electronic apparatus for musical keyboards and specifically to the creation of a tone-generator apparatus and programming method for tuning key selection to the natural scale.
The natural-scale tone-generator apparatus and programming method described herein is designed to create perfect keyboard harmony in any playing key and eliminate the unacceptable distortion created by equal temperament tuning.
The equal-tempered music scale, also called the scale of equal temperment, and more commonly, the tempered scale, was introduced over two hundred years ago to allow fixed-tone instruments such as a piano or an organ to be played in any key by compromising the unequal intervals of the natural scale.
Table 1 on Page 5 compares interval frequencies for one octave of an equal tempered scale of A with the corresponding natural scale of A Major. The column headings of Table 1 are explained as follows:
The "Key" column is for the decimal key number of the tempered-scale and natural-scale frequencies according to a MIDI keyboard Specification. PA1 The "Note" column is for the name of the note corresponding to the tempered-scale and natural-scale frequency. PA1 The "TS Hz" column is for the tempered-scale frequency in Hz. PA1 The "TS Formula" column shows how tempered-scale frequencies are formed by multiplying each successive frequency, starting with a 220.0 Hz reference note, by 1.0595 to get the next frequency. PA1 The "Interval" column shows the name of tempered-scale and natural-scale semitone intervals preceded by their program code numbers in decimal. PA1 The "NS Hz" column is for the natural-scale frequency in Hz. PA1 The "NS Formula" column shows how to calculate the natural-scale frequencies by starting with a reference frequency of 220 Hz for note A and multiplying it by unique whole-number ratios to get the frequencies of successive notes. PA1 The "E-T Error" column shows equal-tempered error in cents. One cent is a 1/100th part of the frequency difference from one semitone to another.
Table 1 permits detailed information about tempered-scale intervals and corresponding natural-scale intervals to be read directly from columns and rows. For example, the scale of A begins with a MIDI key number 57 for a note A and a frequency of 220 Hz, which is also the 1st interval and reference note for each scale and has a TS error of 0.
The next interval is a key number 58 for a note A# with a tempered-scale frequency of 233.1 Hz, calculated by multiplying 220 Hz by 1.0595. This is a Semitone interval and has a natural-scale frequency of 234.7 Hz, calculated by multiplying 220 Hz by a unique whole-number ratio, 16/15, which corresponds to a TS Error of -12 cents. Detailed information for all of the other intervals can be read directly from Table 1 in a similar manner.
In Table 1, the 1st, 2nd, 3rd, 4th, 5th, 6th, 7th, and 8th intervals refer to the original succession of two whole tones and a half tone followed by three whole tones and a half tone discovered by Pythagoras in the 6th century B.C. and called the diatonic or natural scale. The other semitone intervals were added to form the chromatic scale, which allows a musical scale to be started from any one of twelve semitone pitches identified by its key-signature note.
Table 1 shows that the equal-tempered scale is made up of 12 equal semitones to the octave making the frequency ratio of successive semitones the 12th root of 2 or 1:1.0595 approximately. A smaller factor, called the "cent" is used for tuning purposes and is defined as a 1/100th part of an equal-tempered semitone. Human hearing can detect a note that is out of tune within a few cents if there is another note or unison to compare it with. Distortion greater than 3 cents is noticeable to a trained ear.
In contrast with the uniform divisions of the equal-tempered scale, Table 1 shows that the natural scale is made up of a series of unequal semitone intervals created by small, whole-number ratios. These ratios were also discovered by the Greek philosopher, Pythagoras, which he calculated from the lengths of strings on musical instruments corresponding to the most pleasant sound combinations. They remain the basis of all harmony today.
Because the whole tones of the natural scale are unequal in size, the twelve musical scales starting with each semitone of the natural scale produce some notes that are too far out of tune for use in other playing keys. The tempered scale equalizes semitone sizes so that they can be used in all playing keys, but the distortion is still at an unacceptable level for nine of the twelve semitone intervals as shown in Table 1.
The key numbers in Table 1 are decimal values of MIDI hexadecimal key numbers used to identify key selection according to a binary code defined by the Musical Instrument Digital Interface Specification, which is more fully explained on Page 7. Commonly known as the MIDI Specification, a copy is available on the internet, in the MIDI literature of public libraries, by manufacturers, and in publications of music-trade magazines. An excellent reference for the mechanics of the musical scale is a book called PIANO TUNING and ALLIED ARTS, by William Braid White, Mus. D., and published by the TUNERS SUPPLY COMPANY of Boston, Mass.
The chord of A Major, which consists of the 1st, 3rd, and 5th intervals, can be used to illustrate the harmonic distortion created by the tempered scale. Table 1 shows that the natural-scale frequency of the 3rd interval, note C#, is 275.0 Hz and that the corresponding frequency in equal temperment is 277.2 Hz, which is 2.2 Hz higher than it should be. In equal temperment, the number of hertz from C# to C is 15.6 (277.2-261.6) making the 3rd interval 14 cents too sharp (100.times.2.2/15.6) as shown in the TS Error column.
The chord of A Minor 6th, which is made up of the 1st, Minor 3rd, 5th, and 6th intervals, is even more discordant. In equal temperment, Table 1 shows that notes C and F# are 32 cents out of tune with one another.
As it usually requires a 3rd or Minor 3rd interval to form a chord, it is thus apparent that all chords in equal temperment contain notes that can be out of tune up to 10 times the maximum tolerance level (3 cents) for good harmony.
TABLE 1 __________________________________________________________________________ Comparative Frequencies of Equal Tempered Scale of A and Natural Scale of A Major with corresponding key, note, and interval identification. Table 1 also shows formulas for calculating frequencies and tempered scale error. Key Note TS Hz TS Formula Interval NS Hz NS Formula TS Error __________________________________________________________________________ 57 A 220.0 reference note 0 1st 220.0 reference note 0 Cents 58 A# 233.1 220.0 .times. 1.0595 1 Semtone 234.7 220 .times. 16/15 -12 Cents 59 B 246.9 233.1 .times. 1.0595 2 2nd 247.5 220 .times. 9/8 -4 Cents 60 C 261.6 246.9 .times. 1.0595 3 Min 3rd 264.0 220 .times. 6/5 -16 Cents 61 C# 277.2 261.6 .times. 1.0595 4 3rd 275.0 220 .times. 5/4 +14 Cents 62 D 293.7 277.2 .times. 1.0595 5 4th 293.3 220 .times. 4/3 +2 Cents 63 D# 311.1 293.7 .times. 1.0595 6 Dim 5th 312.9 220 .times. 64/45 -10 Cents 64 E 329.6 311.1 .times. 1.0595 7 5th 330.0 220 .times. 3/2 -2 Cents 65 F 349.2 329.6 .times. 1.0595 8 Min 6th 352.0 220 .times. 8/5 -14 Cents 66 F# 370.0 349.2 .times. 1.0505 9 6th 366.7 220 .times. 5/3 +16 Cents 67 G 392.0 370.0 .times. 1.0595 10 Min 7th 391.1 220 .times. 16/9 +4 Cents 68 G# 415.3 392.0 .times. 1.0595 11 7th 412.2 220 .times. 15/8 +12 Cents 69 A 440.0 415.3 .times. 1.0595 0 8th 440.0 220 .times. 2/1 0 Cents __________________________________________________________________________
As well as having whole tones of unequal size, it is also true that natural scale octaves derived by the circle-of-fifths method of Pythagoras are not exact multiples of each other. The invention solves these problems by using the tempered scale as a reference to keep octaves coincident and tuning corrections equal for the same interval in each of the 12 semitone playing keys.
A unique sequence of computer-program steps can then reduce key selection to intervals of one octave and correct tone-generator apparatus from the tempered scale to the natural scale with a simple look-up table of 12 correction codes.
The invention apparatus is therefore connected to get key selection data from the binary code of a MIDI keyboard or key sequencer and to control tuning of tone generator apparatus with a computer and computer program.
The tuning program makes an on-going, note-on list of key selection. The list is then converted to intervals of one octave for tuning-code identification. The first note in the list then becomes the root position or tempered-scale reference for correcting the rest of the notes in the list to the natural scale. For non-percussive organ voicing, which does not require velocity values, the program substitutes tuning-correction codes for key-velocity values in the MIDI binary code before outputting each note-on key-selection list. As an optional program feature, correction from a current root position could continue as long as the first note in the list is held on. A full explanation of program procedures by flowchart steps are included in the specification.
Perfect tuning is possible using the tempered scale as a reference because, even though the tempered-scale reference frequency may vary from true pitch as much as sixteen cents, it forms the root position for each corrected note and no distortion is detected. As previously stated, without making a comparison, the human ear can't pinpoint a pitch difference, and chords with melody notes are always selected in one semitone playing key at a time.
Tones will sound richer, more spontaneous, and alive with natural-scale corrections constantly changing tempered-scale tuning. The programming method could also be used to access correctly tuned notes of digital algorithms and recordings for more harmonious and less monotonous performance.
In the past, the best tone generators for electronic musical keyboards were assembled from discrete transistor components and associated hardware. Tuning was fixed and was adjusted by turning a slug in a shielded feedback coil. These tone generators, although excellent, were bulky and expensive.
Presently, high global production and the lower cost of semiconductors make phase-locked-loops such as the 567 especially attractive as voltage-controllable oscillators for musical tone generators. These popular integrated circuits are used in many critical-performance applications. As an array of tone generators, they do not interfere with one another and tuning can be controlled by switching voltages digitally with the natural-scale tone-generator program and apparatus.
With the introduction of digital recordings for musical keyboards, it was originally thought that keyboards with live tone generators, such as acoustic strings and electronic oscillators, would soon be a thing of the past. Instead, digital technology would allow computers to generate low cost recordings of musical instrument sounds for keyboard selection. However, after hearing and assessing digitally-recorded keyboard sounds for many years, audiences still prefer a skilled performance on a live instrument and musicians realize that live instruments will always be necessary to develop and maintain musical skills.
Digital music technology has spurred mass production of low cost musical keyboards that can transmit key selection on and off data by means of the MIDI binary code. This information can be stored by computer sound cards for key replay. The natural-scale tone-generator apparatus can be connected to the MIDI output of any MIDI keyboard or sound card as a source of key selection. The MIDI binary code is derived from the MIDI specification and is an internationally accepted binary code for transmitting and receiving key selection and control information. All MIDI communication is propagated by means of multibyte commands consisting of a Status byte followed by one or two Data bytes.
The Status byte is an 8-bit binary number with a most significant bit of one. It serves to define the purpose of the Data bytes that follow the Status byte. There are several kinds of Status bytes in a MIDI transmission and key selection only requires Note-On Status commands to be sorted out for processing.
Data bytes are 8-bit binary numbers with a most significant bit of zero. There are at least two Data bytes following a Note-On command. They identify the key number and key velocity. A key velocity of zero is a Note-Off command. When a Status byte is sent, the receiver remains in that status until a different Status byte is sent. This is called a Running Status and allows long strings of note On-Off messages with only the correct number of Data bytes necessary for transmission or reception. A copy of the MIDI 1.0 Specification is included in books such as MIND OVER MIDI, by the editors of Keyboard Magazine, published by Hal Leonard Corporation, 7777 West Bluemound Rd, Milwaukee, Wis. 53213.